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Abstract

REACT estimators for the mean of a linear model involve three steps: transforming themodel to a canonical form that provides an economical representation of the unknown meanvector, estimating the risks of a class of candidate linear shrinkage estimators, and adaptivelyselecting the candidate estimator that minimizes estimated risk. Applied to one- or higher-way layouts, the REACT method generates automatic scatterplot smoothers that competewell on standard data sets with the best fits obtained by alternative techniques. Historicalprecursors to REACT include nested model selection, ridge regression, and nested principalcomponent selection for the linear model. However, REACT's insistence on working with aneconomical basis greatly increases its superefficiency relative to the least squares fit. Thisreduction in risk and the possible economy of the discrete cosine basis, of the orthogonalpolynomial basis, or of a smooth basis that generalizes the discrete cosine basis are illustratedby fitting scatterplots drawn from the literature. Flexible monotone shrinkage of componentsrather than nested 1-0 shrinkage achieves a secondary decrease in risk that is visible in theseexamples. Pinsker bounds on asymptotic minimax risk for the estimation problem expressthe remarkable role of basis economy in reducing risk